Synthetic Detergents

Soaps are the detergents used since long. Soaps used for cleansing purpose are sodium or potassium salts of long chain fatty acids such as stearic acid, palmitic acid. These cannot be used with hard water. Synthetic detergents are cleansing agents which have all the properties of soap but which actually do not contain any soap. These can be used both in soft water and hard water as they give foam even in hard water. Which is not true for a detergent molecule ?